This document analyzes the kinematic performance of a 4-link planar serial manipulator by determining its condition number from the Jacobian matrix. A computer program in MATLAB was used to calculate the Jacobian matrix, its inverse, and the condition number for varying joint angles and link lengths. The results show that for equal link lengths of 1:1:1:1, the minimum condition numbers obtained were 1.0032 for joint angle 2 of 500°, 1.0473 for joint angle 3 of 400°, and 1.0861 for joint angle 4 of 200°. The condition number measures the accuracy of the end effector velocity and how errors in joint velocities are magnified at the end effector.

1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 179
KINEMATIC PERFORMANCE ANALYSIS OF 4-LINK PLANAR
SERIAL MANIPULATOR
K.Balaji1
, K.Brahma Raju2
, M.Surya Narayana3
1
Asst. Professor, Dept of ME., SIET, Amalapuram, A.P., India
2
Professor & HOD, Dept of ME., S.R.K.R. Engineering College, A.P., India
3
Asst. Professor, Dept of ME., Pragathi Engineering College, A.P. India
Abstract
This investigation is all about determining condition number for four link planner serial manipulator. Condition number which
measures the accuracy of end effector velocity is obtained from the Jacobian matrix of the four link planner serial manipulator.
Inverse of a non-square matrix in the calculation of the condition number. Pseudo inverse procedure is used for the calculation of
inverse of jacobian matrix using condition number. Isotropic condition for manipulator is found. “MAT Lab” 6.1.0 version is used
to determine the condition number which is recently developed for complex matrix manipulations. A computer program in “MAT
Lab” is written for finding Jocobian matrix, inverse of Jacobian matrix, norm of jacobian and norm of Jacobian matrix and the
condition number for various input joint angles and link lengths.
Keywords: Serial manipulator, Condition number, Jacobian matrix, MAT Lab,
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1. INTRODUCTION
Since the industrial Revolution there has been an ever-
increasing demand to improve product quality and reducing
cost. At the beginning of the 20th
century, Ford motor
company introduced the concept of mass production. In
mass production each production machine is designed to
perform a predetermined task, every time a new model is
introduced, the entire production line has o be shut down
and retooled. Retooling production line during each model
change can be very expensive. To overcome the above
problem robot manipulators have been introduced by
manufacturing industries for performing certain production
tasks, such as material handling, spot welding, spray
painting and assembling. As robot manipulators controlled
by computers or microprocessors, they can easily be
reprogrammed for different tasks. Hence there is no need to
replace these machines during a model change. We call this
type of automation as flexible automation.
Robotics is the science of designing and building robots for
real life applications in automated manufacturing and other
non- manufacturing environments. Robots are the means of
performing multi actives for man‟s welfare and integrate
manner, maintain their arm flexibility to do any work
effecting enhanced productivity, guarantying quality,
assuring reliability and ensuring safety to the workers. There
is only one definition of an industrial robot that is
internationally accepted. It was developed by group of
industrial scientists from the robotics industries association
(R.I.A) formerly, Robotics Institute of American 1979.
According to them, “An industrial robot is a
reprogrammable, multi-functional manipulator, designed to
move material, parts, tools or special devices through
variable programmed motions for the performance of a
variety of tasks‟. An industrial robot is a general purpose
programmable machine which possesses certain
anthropomorphic or human like characteristics. The most
typical human like characteristic of present day robots is
their movable arms. The Word the Writer Kernel Capek
introduced in “Robot” to the English language in 1921. In
this work, the robots are machines that resemble people, but
work tirelessly. Among science fiction writers, Isaac
Asimov has contributed a number of stories staring in 1939
and introduced the term “Robotic”.
The seventeenth and eighteenth centuries, there were
number of mechanical devices that had some of the features
of robots. During the late 1940‟s research programs were
started the Oak Ride and Argon national laboratories to
develop remotely controlled mechanical manipulator for the
handling radioactive material. In the mid 1950‟s George C.
Devol developed a device. Further development of this
concept by Devol and Joseph F. Engelbreges led to the
industrial Robot. In the 1960 the flexibility of these
machines were enhanced significantly by the use of sensors
feedback. In the 1970 the Stanford arm a small electrically
powered robot arm, developed at Stanford University. Some
of the most serious work in Robotics began as these arms
were used as robot manipulators. During 1970‟s great deal
of research work focused on use of external sensors to
facilitate manipulative operations. In the 1980‟s several of
the programming system were produced. Today we viewed
robotics as much broader field of work than we did just a
few years ago dealing with research development in a
number of interdisciplinary areas including kinematics,
planning systems, sensing, programming languages and
machine intelligence.

2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 180
Moving the robot joints. Four types of joints used on
commercial robots. The first two types adapted from a
SCARA type robot. The second set a revolute type robot. A
sliding joint which moves along the single Cartesian axis. It
is also known as a linear joint, a Translational joint,
Cartesian joint or a prismatic joint. The remaining joints are
rotational joints. Since they all result in angular movement.
Their chief distinction is the plane of motion that they
support. In the first type, the two links are in different planes
and angular motion is in the plane of the second link (1),
the second type provides rotation in a plane 900
from the line
of either arm link (2), in the case the two links always lie
long a straight line. The third joint is a hinge type (3) that
restricts angular movement to the of both arm links. In this
case the links always stay in the same plane. The
combination of joints determines the type of robots. Most
robot have three or four major joints that define their work
envelope (i.e the area that their end effector can reach)
actions that control wrist motion usually provide only tool
orientation and do not affect work envelope.
2. ROBOT KINEMATICS
Robot kinematics is a study of the geometry of manipulator
arm motions. Since the performance of specific tasks is
achieved through the movement of manipulator arm
linkages, kinematics is a fundament tool in manipulator
design and control. A manipulator consists of a series of
rigid bodies (links) connected by mean of kinematic pairs or
joints. Joints can be essentially of two types; revolute and
prismatic. The whole structure forms a kinematic chain. One
end of the other end allowing manipulation of objects in
space.
A kinematic chain is termed open when there is only
sequence of links connecting the two ends of the chain.
Alternatively, a manipulator contains a closed kinematic
chain when a sequence of link forms a loop. For a robot to
perform a specific task, the location of the end effector
relative to the base should be established first. This is called
position analysis problem. There are two types of position
analysis problems; direct position or direct kinematic and
inverse position or inverse kinematics problems. For direct
kinematic, the joint variables are given and the problem is to
find the location of the end effector. The key to the solution
of the direct kinematics problem was the Denavit-
Hartenberg representation. The relationship between the
joint velocities and the corresponding effector linear and
angular velocities and derived by differential kinematic. This
relation described by matrix, termed geometric Jacobian,
which depends on the manipulator configuration.
3. THE DENAVIT-HARTENBERG
REPRESENTATION
For describing the kinematic relationship between a pair of
adjacent links involved in a kinematic chain. The D-H
notation is a systematic method of describing the kinematic
relationship. The method is based on 4x4 matrix
representation of a rigid body position and orientation. It
uses minimum number of parameters to completely describe
the kinematic relationship.
Fig. Shows a pair of adjacent links; link i-1 and link I and
their associated joint i-1 and i+1; the origin of the ith
frame is
located at the inter section of joint axes i+1 and common
normal between the axes I and i+1. The xi axis is directed
along the extension link of the common normal. While Zi
axis is along the joint axis i+1. Finally the yi axis chosen
such that relative resultant frame oi-xiyizi forms right hand
coordinate systems.
i is the angle between xi-1 and the common normal HiOi
measured about Zi-1 axis in right hand hand sense. Alpha is
the angle between joint axis i and Zi axis. The parameter ai
and αi are constant and they are determined by the geometry
of the link. i and di are varies as the joint moves. Fig shows
two link coordinates frames oi-xiyizi and oi-1-xi-1yi-1zi-1 and
intermediate frame Hi-x‟iy‟iz‟I attached at point Hi.
4. FORMULATION OF JACOBIAN MATRIX
The relationship between the joint velocities and
corresponding end-effector linear and angular velocities are
mapping by a matrix which is known as “Jacobian Matrix”.
Consider n- degree of mobility of manipulator. The direct
kinematic equation can be written in the form
Where q=[q1……………..qn]T
is the vector of joint
variables. If „q‟ varies end effector position and orientation
varies.
The differential kinematics gives the relationship between
joint velocities and end effector linear velocity and angular
velocities. It is desired to express the end effector linear
velocities po
, angular ω as a function of joint velocities qo
by
following relations.

3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 181
Jp is (3xn) Matrix relative to the contribution of joint
velocities qo
to the end effector linear velocities P.
Jo is (3xn) Matrix relative to the contribution of joint
velocities qo
to the end effector angular velocities ω.
The above 1 and 2 equations can be written as
Equation (3) represents the manipulator differential
equation. J is (6xn) matrix is the manipulator geometric
Jacobian matrix.
5. CONDITION NUMBER
Condition number of matrix A is defined as the product of
the norm of A and the norm of A inverse.
Norm of A is given by ||A|| = [trace of (AWAT
)]
Trace is the sum of the matrix.
Condition number of a matrix is always greater or equal to
one. If it is close to one the matrix is well conditioned which
means its inverse can be computed with good accuracy. If
the condition number is large, then the matrix is ill-
conditioned and the consumption of its inverse, or solution
of linear system of equations is prone to large numerical
errors. A matri that is not invariable has condition number
equal to infinity.
Condition number shows how much a small in the input data
can be magnified in the output.
Let Ax = b, then relative solution error is measured by
||$||/||X||=||A|| ||A-1
|| (||$b||/||A||)
Where ||A|| ||A-1
|| = Z, condition number.
$b = small perturbations of b.
$ = deviation of the solution of perturbed system from
original solution.
When condition number criteria is applied to Jacobian
matrix of the velocity transformation equation, =J.
Condition number of Jacobian is given by
Z = ||J|| ||J-1
||
Therefore condition number of a Jacobian matrix gives how
much error in joint velocities magnified into Cartesian
velocities.
As the end effector moves from one location to another
location the condition number varies. The points in the
workspace of a manipulator are called isotropic points ,
when the condition number of matrix is one. Condition
number is the best tool for the control of the velocity of end
effector. Condition number of the accuracy of the end
effector‟s velocity
Jocobian matrix for 4 link planner serial manipulator with
respect to base frame:
Zo, Z1, Z2, Z3, are the unit vectors along the joint axis.
P, Po, P1, P2, P3 are the vectors defined from origin of link
frame.
6. RESULTS AND DISCUSSIONS
Simplified notations are used for easy understanding and
standardized notations are used for finding condition number
and matrices too. Graphs were constructed varying each and
every parameter and graphs were drawn for each and every
table.
For the program it is clear that for link length ratios 1:1:1:1.
The suitable condition number varying joint angle „2‟ is
“1.0032” at 500
. The suitable condition number varying joint
angle „3‟ is “1.0473” at 400
. The suitable condition number
varying joint angle „4‟ is “1.0861” at 200
.
1. The condition number when varying link length 1
”a1” keeping all the terms constant joint „1‟ is 150
, joint
angle „2‟ is 500
, joint angle „3‟ is 130
, and joint angle „4‟
is 610
, and with a2:a3:a4 =1:1:1 is at a1 =1 and the
condition number is “1.0032”.
2. The condition number when varying link length 2
”a2” keeping all the terms constant joint „1‟ is 150
, joint
angle „2‟ is 500
, joint angle „3‟ is 130
, and joint angle „4‟
is 610
, and with a2:a3:a4 =1:1:1 is at a2 =1 and the
condition number is “1.0032”.
3. The condition number when varying link length 3
”a3” keeping all the terms constant joint „1‟ is 150
, joint
angle „2‟ is 500
, joint angle „3‟ is 130
, and joint angle „4‟
is 610
, and with a2:a3:a4 =1:1:1 is at a3 =1 and the
condition number is “1.0032”.
4. The condition number when varying link length 1
”a1” keeping all the terms constant joint „4‟ is 150
, joint
angle „2‟ is 500
, joint angle „4‟ is 130
, and joint angle „4‟

4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 182
is 610
, and with a2:a3:a4 =1:1:1 is at a4 =1 and the
condition number is “1.0032”.
1. Condition number values for joint angles
1= 150
; 2= 500
; 3= 130
; 4= 610
,
Link lengths a1: a2: a3=1:1:1 varying a1.
2. Condition number values for joint angles
1= 150
; 2= 500
; 3= 130
; 4= 610
,
Link lengths a1: a2: a3=1:1:1 varying a2
3. Condition number values for joint angles
1= 150
; 2= 500
; 3= 130
; 4= 610
,
Link lengths a1: a2: a3=1:1:1 varying a3.
4. Condition number values for link length ratio 1:1:1:1
varying joint angle “4”
5. Condition number values for link length ratio 1:1:1:1
varying joint angle “2.”

5. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 183
6. Condition number values for link length ratio 1:1:1:1
varying joint angle “3”.
7. CONCLUSIONS
Condition number for four – link planar serial manipulator
was determined. As the end effector moves from one
location to another, the condition number will vary. It is
noticed that condition number for four-linked planar serial
manipulator is dependent for first joint angle. Condition
number is best tool for the velocity of the end effector.
A program in “MAT LAB” (A high level programming
language for technical computing) was written for finding
condition number and inverse of Jacobian matrix. Condition
number is given by the product of norm of jacobian matrix
and form of inverse of jacobian matrix. It must be strictly
nearer of unity. Best performance of end effector will be
obtained when the condition number reaches unity.
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